City lawyer Assa Nyakundi, who is suspected of killing his son, has not recorded a statement with the police more than a week after the shooting.
Without such a statement, detectives handling the matter are in a dilemma.
Added to the hurdle is the fact that they cannot present him in court or free him as he is being treated at Nairobi Hospital.
All this comes as new information from security cameras appears to contradict Mr Nyakundi's earlier statement that he was near his house in Muthaiga North when the shooting took place last Sunday.
Detectives said there are 31 minutes during which the lawyer's location cannot be accounted for, from the time his car was spotted at Museum Hill to when he arrived in hospital.
Footage from the Police Integrated Command, Control and Communication (IC3) centre shows that the car was captured at Museum Hill at 12.50pm and was later seen entering Aga Khan Hospital from Limuru Road.
It is during the unaccounted 31 minutes that police believe 29-year-old Joseph Nyakundi was killed.
Mr Nyakundi told police that he accidentally shot his son while securing his gun from a holster as they approached his house from church.
The new contradiction is the third inconsistency from the lawyer that police have recorded.
A claim that he shot his son in the car has been contradicted by the postmortem report. It revealed Nyakundi Jr was shot at an angle of 90 degrees.
Pathologists said the bullet was released from outside the car.
The lawyer had said his pistol had 14 bullets, only for the gun to be found with 13 rounds when it was examined.
Meanwhile, the family is not sure when Nyakundi Jr's funeral will be held.
"The matter has been taken up by the Kenya Assemblies of God," said Mr Eliud Nyakundi, the lawyer's younger brother.